Topic 2.9

Logarithmic Expressions

Ever since 2.1 one question has been waiting: when does Nova reach 10,000 subscribers? We built tables and saw that it happens between month 9 and month 10. Now let's set up the equation:

10 MIN READ8 IDEAS33 PROBLEMS15 flashcards

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01
A logarithm is an exponent: $\log_{b} c = a$ ⇔ $b^{a} = c$ .

Practice 33 problems Next: Chapter 2.10 Inverses of Exponential Functions

256\,(1.5)^{t} = 10000 \;\Rightarrow\; (1.5)^{t} = 39.0625 \;\Rightarrow\; t = \log_{1.5} 39.0625

The unknown is an EXPONENT: 1.5 raised to what power gives 39.0625? Addition has subtraction to undo it, multiplication has division, and squaring has square roots. For “what exponent?” we need a new operation, the logarithm.

Remember from 2.8: an inverse runs a function backwards, from output to input. The logarithm runs an exponential function backwards, from the result back to the exponent. (2.10 looks at that inverse relationship as graphs.)

A Logarithm Is an Exponent

CONCEPT

Definition

$\log_{b} c$ (read “log base $b$ of c”; in print the $b$ is written small and low) is the exponent you put on $b$ to get $c$ .

In symbols: $\log_{b} c = a$ means exactly the same thing as $b^{a} = c$ . The base $b$ must be positive and not 1, just like the base of an exponential function (2.3).

Read $\log_{2} 32$ as the question “2 to what power is 32?” The answer is 5, so $\log_{2} 32 = 5$ .

\log_b c = a \quad\Longleftrightarrow\quad b^{a} = c \qquad (b > 0,\; b \ne 1)

Every exponential fact can be rewritten as a logarithm fact, and back. Three parts are involved: the BASE $b$ , the EXPONENT a, and the RESULT $c$ . The base stays the base in both forms. The logarithm always equals the exponent:

exponential form	logarithmic form	in words
$2^{5} = 32$	$\log_{2} 32 = 5$	2 to the 5th is 32
$10^{-2} = 0.01$	$\log_{10} 0.01 = -2$	10 to the −2 is 0.01
$9^{1/2} = 3$	$\log_{9} 3$ = 1/2	the square root of 9 is 3
$b^{0} = 1$	$\log_{b} 1 = 0$	anything to the 0 is 1
$b^{1}$ = b	$\log_{b} b = 1$	anything to the 1st is itself

Worked example

Example 1 (converting). (a) Write $7^{2} = 49$ and $5^{-1} = 0.2$ in log form. (b) Write $\log_{16} 8 = 3/4$ in exponential form.

01
$7^{2} = 49$ : the base is 7, the exponent is 2, the result is 49. The log equals the exponent: $\log_{7} 49 = 2$ .
02
$5^{-1} = 0.2$ : base 5, exponent −1, result 0.2. So $\log_{5} 0.2 = -1$ .
03
$\log_{16} 8 = 3/4$ : base 16, and the $\log(3/4)$ is the exponent, so $16^{3/4} = 8$ . Check: $16^{1/4} = 2$ , and $2^{3} = 8$ ✓.

KEY RULE

A logarithm is an exponent: $\log_{b} c = a$ ⇔ $b^{a} = c$ .

Quick check

Rewrite $\log_4 64 = 3$ in exponential form.

Evaluating Logarithms by Hand

To evaluate $\log_{b} c$ , don't reach for a formula. Ask the question “b to what power is $c$ ?”, and write $c$ as a power of $b$ .

Worked example

Example 2. Evaluate $\log_{3} 81$ , $\log_{10} 1000$ , $\log_{5}(1/25)$ , and $\log_{2}$ $\sqrt{2}$ .

01
$\log_{3} 81$ : 3 to what power is 81? $3^{4} = 81$ , so $\log_{3} 81 = 4$ .
02
$\log_{10} 1000$ : $10^{3} = 1000$ , so $\log_{10} 1000 = 3$ .
03
$\log_{5}(1/25)$ : $1/25 = 1/ 5^{2} = 5^{-2}$ , so $\log_{5}(1/25) = -2$ . A result between 0 and 1 needs a NEGATIVE exponent.
04
$\log_{2}$ $\sqrt{2}$ : $\sqrt{2} = 2^{1/2}$ , so $\log_{2}$ $\sqrt{2} = 1/2$ . Roots give FRACTIONAL exponents (2.4).
When $c$ is not a whole power of $b$ , write both as powers of a smaller common base.

Worked example

Example 3. Evaluate $\log_{4} 8$ and $\log_{16} 8$ .

01
$\log_{4} 8$ : $4 = 2^{2}$ and $8 = 2^{3}$ . Set up $4^{a} = 8$ and rewrite with base 2:
$\log_4 8 = a \;\Rightarrow\; 4^{a} = 8 \;\Rightarrow\; (2^{2})^{a} = 2^{3} \;\Rightarrow\; 2a = 3 \;\Rightarrow\; a = \tfrac{3}{2}$
02
Check: $4^{3/2} = (\sqrt{4})^{3} = 2^{3} = 8$ ✓.
03
$\log_{16} 8$ , the same way:
$\log_{16} 8 = a \;\Rightarrow\; 16^{a} = 8 \;\Rightarrow\; (2^{4})^{a} = 2^{3} \;\Rightarrow\; 4a = 3 \;\Rightarrow\; a = \tfrac{3}{4}$

COMMON MISTAKE

“ $\log_{2} 8 = 8 \div 2 = 4$ .”

A logarithm is not a division. $\log_{2} 8$ asks for the exponent: $2^{3} = 8$ , so $\log_{2} 8 = 3$ . Check any answer by raising the base to it: $2^{4} = 16$ , not 8.

“ $\log_{4} 8 = 2$ , because 8 is $2 \times 4$ .”

Again, check by raising: $4^{2} = 16 \ne 8$ . The right answer is $3/2$ .

Worked example

Try it yourself. Evaluate without a calculator: (a) $\log_{2}(1/8)$ (b) $\log_{27} 3$ (c) $\log_{10} 0.001$ (d) $\log_{6}(1/36)$ (e) $\log_{9} 27$

Answers: (a) −3, because $2^{-3} = 1/8$ . (b) $1/3$ , because $27^{1/3} = 3$ . (c) −3, because $10^{-3} = 0.001$ . (d) −2. (e) $3/2$ , because $9^{3/2} = 3^{3} = 27$ .

Logarithms Between Whole Numbers

Most logarithms aren't whole numbers. You can still pin them down by finding the powers on either side.

Worked example

Example 4. Between which two whole numbers is $\log_{2} 20$ ? And $\log_{10} 350$ ? And $\log_{1.5} 39.0625$ ?

01
$2^{4} = 16$ and $2^{5} = 32$ . Since $16 < 20 < 32$ , $\log_{2} 20$ is between 4 and 5. (A calculator gives about 4.3219.)
02
$10^{2} = 100$ and $10^{3} = 1000$ , so $\log_{10} 350$ is between 2 and 3. (About 2.5441.)
03
Nova: $1.5^{9} \approx 38.44$ and $1.5^{10} \approx 57.67$ , and $38.44 < 39.0625 < 57.67$ . So $\log_{1.5} 39.0625$ is between 9 and 10, about 9.04. That matches the table from 2.1: Nova passes 10,000 early in month 10.
The logarithm measures position on a scale where each step MULTIPLIES instead of adds. On an ordinary number line, 1, 2, 4 and 8 crowd together; on a base-2 log scale they are equally spaced:

On the log scale, the position of each number is its base-2 logarithm.

The Two Logarithms on Your Calculator

CONCEPT

Common Log and Natural Log

The common logarithm has base 10 and is written without a base: $\log x$ means $\log_{10} x$ . ( $\log 1000 = 3$ , $\log 0.1 = -1$ .)

The natural logarithm has base $e \approx 2.718$ (2.3) and has its own name: $\ln x$ means $\log_{e} x$ .

Your calculator has a LOG key and an LN key. For other bases, like $\log_{1.5}$ , you'll learn a trick in 2.12.

Worked example

Example 5. Evaluate $\ln e^{5}$ , $\ln(1/e)$ , ln $\sqrt{e}$ , and $\ln 20$ .

01
$\ln e^{5} = 5$ , because $e^{5}$ is $e$ to the 5th. (ln just reads off the exponent of $e$ .)
02
$\ln(1/e) = \ln e^{-1} = -1$ , and ln $\sqrt{e} = \ln e^{1/2} = 1/2$ .
03
$\ln 20$ isn't a simple power of $e$ : use the LN key. $\ln 20 \approx 2.9957$ , so $e^{2.9957} \approx 20$ .

COMMON MISTAKE

“ $\log 10$ and $\ln 10$ are the same thing.”

They have different bases. $\log 10 = 1$ , because $10^{1} = 10$ . $\ln 10 \approx 2.3026$ , because $e \approx 2.718$ needs a bigger exponent to reach 10. Always check which key the problem means.

Quick check

Evaluate $\log_2 40$ to four decimal places.

What Logarithms Can't Do

Remember from 2.3: $b^{x} > 0$ for every $x$ . A positive base raised to ANY power is positive. So no exponent turns $b$ into 0 or into a negative number.

CONCEPT

The Input of a Logarithm

$\log_{b} c$ is defined only for $c > 0$ . $\log 0$ and $\log(-100)$ are undefined.

For $0 < c < 1$ , the logarithm is negative (a negative exponent makes a fraction): $\log 0.5$ is between −1 and 0. For $c > 1$ (with $b > 1$ ), it is positive. For $c = 1$ , it is 0.

COMMON MISTAKE

“ $\log(-100) = -2$ .”

$10^{-2} = 0.01$ , not −100. A NEGATIVE logarithm means a small positive input, not a negative input. No power of 10 is negative, so $\log(-100)$ is undefined.

Logarithmic Scales in Real Life

Some quantities range over huge sizes: the ground motion of earthquakes can differ by factors of millions. Scientists report them with a logarithm, so the numbers stay small and each step up means “multiplied by 10”.

REAL-LIFE EXAMPLE

Earthquake Magnitude

The magnitude $M$ of an earthquake is based on the logarithm (base 10) of how strongly the ground shakes. Going up 1 in magnitude means the ground motion is 10 times bigger.

A magnitude 7.0 quake versus a 5.0 quake: 2 steps, so $10^{2} = 100$ times the ground motion, not “2 more”. A 6.5 versus a 6.0: half a step, $10^{0.5} \approx 3.16$ times.

Equal steps on the magnitude scale are equal FACTORS of ground motion.

Practice

Worked example

P1. Rewrite $6^{3} = 216$ in logarithmic form, and $\log_{8} 4 = a$ in exponential form. Find a.

Answer: $\log_{6} 216 = 3$ . And $8^{a} = 4$ : $2^{3a} = 2^{2}$ , so $a = 2/3$ .

Worked example

P2. Which is larger, $\log_{2} 30$ or $\log_{3} 30$ ? Explain without a calculator.

Answer: $16 < 30 < 32$ puts $\log_{2} 30$ between 4 and 5; $27 < 30 < 81$ puts $\log_{3} 30$ between 3 and 4. So $\log_{2} 30$ is larger (about 4.907 vs 3.096). A smaller base needs a larger exponent.

Worked example

P3. How many times stronger is the ground motion of a magnitude 7.2 earthquake than a 5.2?

Answer: 2 steps: $10^{2} = 100$ times.

Worked example

P4. Evaluate $\ln(e^{-2})$ and $\log_{3}(1/\sqrt{3})$ , and find the two whole numbers that $\log_{3} 50$ lies between.

Answer: $\ln(e^{-2}) = -2$ . $1/\sqrt{3} = 3^{-1/2}$ , so $\log_{3}(1/\sqrt{3}) = -1/2$ . Since $27 < 50 < 81$ , $\log_{3} 50$ is between 3 and 4.

Common slips

“ $\log_{2} 8 = 8 \div 2 = 4$ .”

A logarithm is not a division. $\log_{2} 8$ asks for the exponent: $2^{3} = 8$ , so $\log_{2} 8 = 3$ . Check any answer by raising the base to it: $2^{4} = 16$ , not 8.

“ $\log_{4} 8 = 2$ , because 8 is $2 \times 4$ .”

Again, check by raising: $4^{2} = 16 \ne 8$ . The right answer is $3/2$ .
“ $\log 10$ and $\ln 10$ are the same thing.”

They have different bases. $\log 10 = 1$ , because $10^{1} = 10$ . $\ln 10 \approx 2.3026$ , because $e \approx 2.718$ needs a bigger exponent to reach 10. Always check which key the problem means.
“ $\log(-100) = -2$ .”

$10^{-2} = 0.01$ , not −100. A negative logarithm means a small positive input, not a negative input. No power of 10 is negative, so $\log(-100)$ is undefined.

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15 cards · Logarithms, Log or exponential graph?

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Recap card

6 lines to re-read the night before.

01
A logarithm is an exponent: $\log_{b} c = a$ ⇔ $b^{a} = c$ ( $b > 0$ , $b \ne 1$ ). $\log_{b} 1 = 0$ and $\log_{b} b = 1$ .
02
To evaluate by hand, write $c$ as a power of $b$ (use a common base like 2 if needed; roots are fractional exponents). Check by raising the base to your answer.
03
Logs between whole numbers: find the powers of $b$ on either side.
04
$\log x$ means base 10; $\ln x$ means base $e$ , and $\ln e^{k} = k$ .
05
$\log_{b} c$ exists only for $c > 0$ . Inputs between 0 and 1 give negative logs.
06
Log scales turn equal steps into equal factors (earthquakes: +1 magnitude = ×10). Next, in 2.10: the graph of $y = \log_{b} x$ .

Practice 33 problems Next: Chapter 2.10 Inverses of Exponential Functions

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Logarithmic Expressions

A Logarithm Is an Exponent

Evaluating Logarithms by Hand

Logarithms Between Whole Numbers

The Two Logarithms on Your Calculator

What Logarithms Can't Do

Logarithmic Scales in Real Life

Practice

Recap card

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